The EV Formula and the EV of a Call

Assumptions: All examples assume 100bb effective stacks in a 6-max online cash game with no rake, unless a different stack size is stated in the example.

Every poker decision you will ever make can be graded with one number: its expected value. Not whether it won the pot. Not whether it felt right. EV — the average amount you win or lose by taking that action, if you could replay the exact same spot thousands of times.

This lesson builds the formula from scratch and applies it to the cleanest possible case: calling a bet on the river, where there are no future streets to worry about. Once you can compute EV(call) cold, every harder calculation in this module is just the same idea with more branches.

The formula

Expected value is a weighted average. List every way the hand can end, attach a probability to each outcome, attach a payoff to each outcome, multiply, and add it all up:

EV = Σ (probability of outcome × payoff of outcome)

Three rules make this work in practice:

• The probabilities must cover everything that can happen and sum to 100%.
• Payoffs are measured from the moment of the decision. Money already in the pot is not yours anymore — it belongs to the pot. What matters is what you can win from here and what you must risk from here.
• The result is an average, not a prediction. A +4bb call still loses most of the time if your win probability is 35%. It is +4bb because the wins are big enough to pay for all the losses, on average.

That last point is where beginners go wrong. EV does not tell you what will happen this hand. It tells you what the decision is worth. A profitable call that loses was still a profitable call.

EV of a river call

The river call is the simplest real decision in poker because there are exactly two outcomes: you win the pot plus the bet, or you lose your call. No redraws, no future bets, no position to worry about.

EV(call) = (win% × amount you win) − (lose% × amount you lose)

• Amount you win = the pot before the bet, plus the opponent's bet
• Amount you lose = your call

Let's put real cards on it.

BB

A♥♥A♥

10♣♣10♣

BTNunknown

Flop

K♦♦K♦

Turn

You defend the big blind with A♥T♣ and reach the river on K♦T♦6♠4♣2♥ with about 20bb in the middle. The button bets 10bb. Your read: this player barrels his missed diamond draws and worse tens often enough that you expect to win roughly 35% of the time you call.

Plug it in:

• Win: you collect the 20bb pot plus his 10bb bet = 30bb, and this happens 35% of the time
• Lose: you lose your 10bb call, 65% of the time

EV(call) = 0.35 × 30 − 0.65 × 10 = 10.5 − 6.5 = +4bb

Every time you make this call, you earn 4bb on average. Over a session you will mostly remember the times Kx tabled and you mucked — that's the 65%. The 35% pays for all of it and then some.

The same call with worse equity

Now change one number. Suppose this opponent almost never bluffs the river — he bets his draws on the flop and turn but gives up when they miss. Your win estimate drops to 25%:

EV(call) = 0.25 × 30 − 0.75 × 10 = 7.5 − 7.5 = 0bb

Exactly breakeven. Calling and folding are worth the same; the call wins just often enough to pay for itself and not a chip more.

Drop the estimate to 20% — he's only betting hands that beat you plus the very occasional airball:

EV(call) = 0.20 × 30 − 0.80 × 10 = 6 − 8 = −2bb

Same cards, same bet, same pot. The only thing that changed is your estimate of how often you win, and the call swung from clearly profitable to a 2bb mistake. This is the entire skill of river calling: the formula is trivial, and the work is in estimating win%. Hand reading feeds the number; EV grades it.

Breakeven equity is just pot odds

Notice where the call flipped: at exactly 25%. That is not a coincidence. The breakeven point of EV(call) is the win% where the formula equals zero:

win% × (pot + bet) = lose% × call

Solve it and you get the pot odds formula you already know:

required equity = call ÷ (pot + bet + call)

For our spot: 10 ÷ (20 + 10 + 10) = 10/40 = 25%. Facing 10 into 20, you're calling 10 to win 30, which is 3:1 — and 3:1 means you need to win one time in four.

So pot odds were never a separate concept. "Do I have the right pot odds?" is just shorthand for "is EV(call) above zero?" The EV formula is strictly more useful, though, because it doesn't just tell you whether the call is good — it tells you how good. A call that needs 25% and wins 35% is worth +4bb. A call that needs 25% and wins 26% is technically correct and worth about 30 cents at $1 blinds. Knowing the difference tells you which spots deserve your attention and which are close enough that either choice is fine.

EV of an all-in call

The same two-outcome math applies any time a call ends the action — including calling a shove on the flop. There are still cards to come, but because nobody can bet again, your equity in the pot is the whole story.

BTN

8♣♣8♣

8♦♦8♦

UTGunknown

Flop

7♥♥7♥

Turn

This example deviates from the module default: villain sits on a 50bb stack. He opens to 3bb from UTG, you 3-bet 8♣8♦ to 9bb on the button, he calls. The pot is 19.5bb. The flop comes 7♥5♠2♦ and he rips in his last 41bb.

First, the price. You call 41bb to win the 19.5bb pot plus both stacks' flop money — 19.5 + 41 + 41 = 101.5bb total. Required equity = 41 ÷ 101.5 = 40%.

Second, your equity. Say his jam range here is heavy on unimproved big cards like A♥K♥ trying to take the pot down. Against exactly A♥K♥ on this board, 8♣8♦ has 70% equity (computed by simulation — two overcards with backdoor draws pull in about 30%).

Now the EV, with one twist: this time "win%" is your equity, because the all-in freezes the betting and the remaining cards simply run out.

EV(call) = 0.70 × 101.5 − 41 ≈ 71 − 41 = +30bb

Note the form: equity × (total pot after your call) − your call. It's the same formula as the river call, just written more compactly — winning "pot plus bet" after committing your call is identical to winning the whole final pot minus the call you put in. Use whichever framing keeps your arithmetic straight; they always agree.

A +30bb call is enormous — nearly a third of a buy-in earned the moment you click call, regardless of which 30% of the time the ace or king spikes. When it does spike, you did not "get unlucky and make a mistake." You made +30bb and variance took the pot. Those are different things, and the variance lesson later in this module deals with keeping them separate in your head.

Where does the 35% come from?

Fair question: the formula is exact, but we fed it an estimated win rate. That estimate isn't a vibe — it's a count. Go back to the A♥T♣ hand and think about what the button's 10bb river bet actually represents. He bets some set of value hands (Kx, slowplayed sets, the occasional rivered two pair) and some set of bluffs (the missed diamond draws, busted gutshots like QJ). Your AT beats essentially none of the value and all of the bluffs, so your win% is just the bluff share of his betting range.

Suppose your hand reading puts him on roughly 20 value combos and 11 bluff combos in this spot. Then you win 11 of every 31 calls — 11/31 ≈ 35%. That's where the number came from. Change the read to 25 value combos and 6 bluffs, and you win 6/31 ≈ 19% — below the 25% requirement, and the call flips to a fold. The combinatorics module covers the counting machinery in detail; what matters here is the division of labor. The EV formula converts a read into a decision. It cannot fix a bad read. Players who get this backwards obsess over decimal places in the arithmetic while plugging in win rates they invented from frustration ("he's always bluffing"). Spend your effort where the uncertainty lives: the inputs.

This is also why the same bet from two different opponents produces two different correct answers. Versus the barrel-happy regular, 35% is real and the call prints. Versus the passive player who only bets the river when he has it, your true win% might be 10% and the identical call burns 5bb. The formula didn't change. The input did.

Doing it at the table

You will not run three-decimal multiplication in a 20-second time bank. You don't need to. The practical workflow:

1. Price first. Compute required equity from the bet and pot — call ÷ (pot + bet + call). Facing half pot you need 25%, two-thirds pot needs about 29%, full pot needs 33%. These you memorize.
2. Estimate win%. This is hand reading: what does he bet here, and how much of it do you beat?
3. Compare. Win% above the requirement → calling is +EV. The bigger the gap, the more profitable the call.

Then, away from the table, do the full EV math on the hands that felt close. Computing exact EVs in review is what calibrates your in-game estimates — after fifty reviewed river spots, "this feels like 35%" starts being right.

One habit to build now: when a call loses, re-grade it by the numbers that were available at the time, not by the card that hit. EV(call) = 0.35 × 30 − 0.65 × 10 = +4bb is true before, during, and after villain tables a king.

Common errors

• Counting your own past bets as winnable. The pot you can win includes money you put in on earlier streets, sure — but it's pot money now, the same as anyone's. Never "call because I've already invested so much." The formula only sees what you risk and what you can collect from this point forward.
• Using equity where you need win%. On the river there are no cards to come; "equity" just means the fraction of his betting range you beat. Before the river, equity from a calculator only equals your win% when the call is all-in. If betting continues on later streets, raw equity overstates or understates the truth — that gap is the EV-versus-equity lesson later in this module.
• Treating breakeven as a target. Needing 25% and having 26% means the call is barely better than folding. Fine — take it. But don't congratulate yourself; the money in poker comes from spots where the gap is wide, like 70% equity against a 40% price.
• Forgetting the lose term. EV has two halves. 0.35 × 30 is not the EV of the call; it's the gross win rate. Beginners who skip the −0.65 × 10 term think every call is profitable.

The formula scales from here. A bluff is the same equation with fold% in place of win%. A value bet is two EVs compared side by side. A full decision tree is just several of these stacked together. Master the two-outcome version first — probability times payoff, summed — and the rest of this module is bookkeeping.